J. Phys. Chem. 1995,99, 10797-10801
10797
Mean Energy Transfer within the Sequential Direct Encounter Model: Temperature Dependence of (AE) Elena I. Dashevskaya, Evgueni E. Nikitin,*?+and Izhack OrePC Department of Chemistry, Technion, Israel Institute of Technology, Haifa 32000, Israel Received: May 17, 1994; In Final Form: March 30, 1995@
The model of energy transfer in collisions of atoms with vibrationally excited polyatomic molecules suggested recently by the present authors (J. Phys. Chem. 1993, 97, 9397) and based on consecutive direct encounters is generalized. Incorporation of the lifetime distribution of collisions into the Landau-Teller model of the description of the energy transfer in a single encounter yields a general formula for the temperature dependence of the mean energy, (A@, transferred in a collision. This formula takes into account the interplay between the positive temperature dependence originating from the energy-gap law and the negative temperature dependence which is related to the collision lifetime. Good agreement of the temperature dependence of (A@ between the present model and trajectory calculations is obtained.
1. Introduction Recently, we have suggested a model for energy exchange in thermal collisions between a noble gas atom and a polyatomic highly vibrationally excited molecule.’ The model is based on the idea that when a collision complex is formed, the energy transfer occurs in a sequence of direct encounters of a heat bath atom with individual atoms of a molecule. Qualitatively, the result of each encounter was described by an approach similar to that suggested first by Landau and Teller*with a modification which takes into account the acceleration of the colliding particles due to the attractive part of the potential. The mean energy transferred during the lifetime of the collision complex, (AE), was related to the averaged (over encounters) meansquared energy transferred in a direct collision (A@&) via an equation similar to the relaxation equation derived by Landau and Teller.* In this way, it was possible to correlate experimental values of (AE) for azulene-noble gas collisions with those parameters of the colliding partners that determine a single-encounter outcome, It was shown in ref 1 that the leveling off of the curve of the dependence of (AE) on the noble-gas atomic number M is due to the interplay of two effects: the increase in the interaction strength with increase of M and the decrease of the energy-transfer efficiency due to the adiabaticity restrictions resulting from longer collision time for heavy noble gases. The restrictions on energy transfer imposed by the conditions of a nearly adiabatic regime of collisions are usually associated with the so-called energy-gap law. In turn,the manifestation of the energy-gap law is usually accompanied by the positive temperature dependence of the relevant rate coefficient. This feature of the collisional energy-transfer process is due to the fact that both properties, the dependence of the rate coefficient on the energy transferred and on the time of the collision in which this transfer occurs, are described by a single exponential function which reflects the Ehrenfest adiabatic principle. On the other hand, there are other sources of the temperature dependence of the energy-transfer rate coefficient, which are not directly related to the adiabatic restrictions. Available experimental data on (AE) indicate a very diverse temperature dependence of this quantity for different partners,
(e&).
’
A corresponding member of the Institute of Chemical Physics, Russian Academy of Sciences, Moscow. Abstract published in Advance ACS Abstracts, June 1, 1995. @
and very often it is n e g a t i ~ e .Therefore, ~ a question arises as to what extent the energy-gap law is compatible with the negative temperature dependence of (AE). We will answer this question within the sequential direct encounter (SDE)model, which takes into account the formation of a chattering collisions that lives long enough to encompass several direct encounters of a heat-bath atom with atoms of the molecule. The basic idea of the SDE model is in line with recent numerical studies of the mechanisms of the energy flow in collision of triatomictriatomic: diatomic-diat~mic,~and atom-polyatomic6-* molecules. The latest results of quasiclassical trajectory calculations on the Ar-toluene system8indicate that four types of collisions appear to exist. (i) Direct Collisions. These are collision which last less than 300 fs and comprise a single encounter of an heat-bath atom with an atom of the molecule. Some of these encounters are impulsive, and the range of vibrational periods covered by them corresponds to the molecular vibrations below 300 cm-I. These values are (barely) within the range of the lowest-lying out-ofplane modes (ca. 400 cm-I) and within the C-C bends (ca. 300 cm-I). It is therefore possible for the atom to be “kicked” in a single vibration. We emphasize, however, that not all the direct collisions are impulsive, and it is the deviation from the impulsive regime that imposes the adiabaticity restriction on the energy transfer. The direct collisions make up the majority of the inelastic collisions. About 60% of all argon-toluene inelastic collisions are direct, and they transfer ca. 50% of the energy. (ii) Chattering Collisions. These collisions last longer than a single encounter but are too short to allow a complete intramolecular energy redistribution, IVR,to take place during the collision lifetime. A chattering collision does not mean that the atom is located in one place and exercises a vibrational motion. Rather, a detailed analysis of single trajectories indicate that it hops from one site to another sampling part of the minima in the atom-molecule multidimensional potential surface. Chattering collisions comprise the second largest group of collisions after direct collisions. Judging from trajectory calculation results, no less than 30% of all inelastic collisions are chattering collisions. (iii) Complex-Forming Collisions. A complex-forming collision is a collision with a lifetime longer than IVR lifetimes. In such a complex, energy is distributed statistically among all
0022-3654/95/2099-10797$09.00/0 0 1995 American Chemical Society
Dashevskaya et al.
10798 J. Phys. Chem., Vol. 99, No. 27, 1995
the modes of the complex and RRKM theory can be used to calculate rate coefficients for the dissociation of the complex and the kinetic energy release of the fragments. Trajectory calculation results show that for the Ar-toluene pair and the number of these collisions is very small, and they hardly contribute to the energy-transfer process. (iv) Supercollisions. These are collisions which transfer an inordinate amount of energy in a single event. They comprise about 0.1% of all collisions and will not be treated under the present model. In view of the complicated nature of the collision dynamics, there is little hope of developing a comprehensive model for a general case. Therefore, any additional test, by using either experimental data or exact theoretical results, is of supreme importance. In this respect, we consider the ability of a sequential collision model to qualitatively explain’ the “unusual” behavior of (AE) vs M behavior9 to be very promising, though the quantitative description was achieved by introducing an additional parameter Rmin.We emphasize that the value of this parameter which represents the frequency of a “relaxing mode” is of the same order of magnitude that can be expected from the combinational normal-mode frequencies of the excited polyatomic molecule. An additional test of the model might be the verification of the temperature dependence of (AE) and ( A @ ) and the study of the dependence of these quantities on the vibrational energy E of the molecule. This requires, of course, a more detailed formulation of the model with respect to the nature of the chattering collisions. This is the aim of the present study.
2. The Model We reiterate here the main assumptions of the SDE model’ and supplement them with an additional assumption which will make it possible to incorporate chattering collisions of arbitrary lifetime. 1. The model incorporates three characteristic times: i. The inverse frequency 1/Qk of a “relaxing mode”. The main reason for distinguishing between a relaxing mode and an unrelaxing mode is the well-established fact that highfrequency modes virtually do not participate in the energy exchange during the collision. Of course, the energy is shared statistically between all the modes during the time between collisions. ii. The time of a direct encounter, z. The encounter time z is related to the logarithmic derivative of the repulsive part of the interparticle potential a and to the collision velocity v of an impinging atom prior to its arrival at the repulsive wall of the potential.’ The product zQk characterizes the efficiency of the energy exchange for the kth mode. If zQk >> 1, the collisional energy exchange with this mode is very inefficient (adiabatic principle restriction). ... 111. The mean lifetime of the chattering collision, 7., A chattering collision is defined by the condition that the interaction energy at the attractive part of the potential exceeds (in its absolute value) the initial mean kinetic energy. Within the model under discussion, the chattering collision is formed prior to energy exchange with the vibrational energy of the molecule. It is formed because the radial relative translational energy is changed as a result of the energy exchange between the translational and rotational motion of the colliding partners, yith the possible participation of the transitional modes. Since tc > z, and therefore Q;zkc > Qkz, the long-range part of the potential is not important in the energy exchange unless the product Q,; is very small, ~ k > 1. The mean energy transferred is
(hE) = (l/Y)(Ei - Eeq)
(15)
The special case of the strong-collision limit corresponds to the statistical limit when the heat capacity of accepting modes approaches or even exceeds the heat capacity of the molecule, i.e., y 1. In this (rather unrealistic) case the mean energy transferred is
-
As discussed in our previous paper,’ the collisions of noblegas atoms with polyatomic molecules fall in the weak-collision category. Therefore eq 14 applies. If the initial vibrational energy, E, strongly exceeds the mean thermal energy, this equation simplifies to
Dashevskaya et al.
10800 J. Phys. Chem., Vol. 99, No. 27, 1995 The temperature dependence of (A@ enters the eq 17 explicitly via the factor ( k u P 2 and implicitly via 7, and (Ai?d,r)eq. We discuss this implicit dependence in the next section.
4. Temperature Dependence of
(AE2dir), (AE’dir)eq,
and
Within the first-order perturbation approach, the temperature dependence of (hE2d1,) can be found from the work done by an external force on the oscillator. According to eq 3 from ref 1, (AE2d,,>=
Ww*~)’ exp(-Q,,,z)
(18)
+
where E* = kT 60 and z = (1/a)(2E*/p)-II2, with EO being the potential well depth. An additional factor F(E) comes from the average kinetic energy of the relaxing mode; according to ref 2, it will be taken proportional to E. Collecting all contributions which depend on T and the reduced mass p , we arrive at the expression
where TB is the well depth of the atom-molecule potential in Kelvins, TB = cdk. According to this expression, (hE2dl,) increases with temperature. In the limit of impulsive collisions, when the exponent is small, (A@& increases linearly with (T TB) (for a fixed value of E ) . If, in addition, the potential is shallow, TB > 1, and the preexponential factor in eq 24 becomes unity. This limit corresponds to the relaxation of diatomic molecules with the repulsive atom-molecule interaction; in this case g is usually very large, and the exponent (g/8)l/* is also large. Finally, a direct impulsive collision corresponds to m = 1, g = 0. In general, (AE), as defined by eq 24, exhibits non-monotonic temperature dependence, passing through a minimum. The turnover reduced temperature 8*, which separates regions on the negative and positive temperature dependence of (AE), is found from the equation
The solution of this equation reads
O* = 2m2/g
+ [4m4/g2+ 4m2/gl1‘’
(26)
This expression answers the question whether the manifestation of the energy-gap law in the mean energy transferred, AE, is compatible with with negative temperature dependence of AE: For T < 8*TB, the temperature dependence of (AE) is negative, and for T > O*TB, it is positive.
5. Discussion and Conclusion Trajectory calculations of energy transfer between an atom and a polyatomic molecule reveal a wide distribution of the collision events over the lifetimes of a collision c ~ m p l e x .If’ ~ supercollisions are excluded from the discussion (see above), the other types of collisions can be treated, within a certain approximation, in the framework of the sequential direct encounter model. The model suggested takes into account the possibility of formation of “complexes” of different lifetime and
The Temperature Dependence of (AE)
J. Phys. Chem., Vol. 99, No. 27, 1995 10801
the energy exchange in direct encounters between heat-bath atom and atoms of a polyatomic molecule. The reasonable assumption that the exponential function governs the lifetime distribution of “complexes” allows one to arrive at a simple formula which expresses the mean energy transferred as a function of the heat-bath temperature T,depth of the potential well of the atom-molecule intraction expressed through the temperature TB, effective number of degrees of freedom of the collision moiety m, and the energy-gap temperature To. According to the model adopted, TB for a given collision partner is known and can be expressed via the boiling temperatures of pure substances.’ The temperature TG contains the reduced mass of the colliding pair and the combination of two unknown parameters, the steepness of the repulsive branch of the potential a and the relaxing mode frequency !&. One can only suggest a reasonable guess for the values of these parameters.’ On the other hand, Tgcan be extracted from the experimental data for (A@ measured at the same temperature but for different noble gases. Once TG is known, eq 23 yields the temperature dependence of (A,??) provided one adopts a value of m. Within the simple picture of the intermediate complex one can take m = 3 (one stretching and two bending modes of an atom attached to the molecule). A stringent test of the model would be the case when TB, TG,and m could be estimated a priori and not be considered as fitting parameters. This is the case for He as a collider, at not too low temperatures. In this case, the binding energy is very small (TB/T-=x1) and the collisions are impulsive (T,/T a 1). Then, according to eqs 19 and 23, we have
(AE) = constant
(27)
Recent trajectory calculations by Lim20for the toluene-helium system give the following temperature-dependent relationships:
(&)
oc
@0.88f0.20)
Clearly, there is a very good agreement between trajectory calculations and predictions of the model.
In conclusion, we suggest that eq 24 can serve as a basis for the correlation of the experimental data on (AE) for a given polyatomic molecule at a given degree of internal excitation. The experiments should be carried out for different noble-gas collision partners and at different heat-bath temperatures. Our model does not provide estimates of the absolute values of the energy transferred since this would require more detailed information on the collision dynamics compared to that built in the model. Acknowledgment. This research is supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel, and by the Fund for Promotion of Research at the Technion. References and Notes (1) Dashevskaya, E. I.; Nikitin, E. E.; Oref, I. J. Phys. Chem. 1993, 98, 9397. (2) Landau, L.; Teller, E. Phys. Z. Sow. 1936, 10, 34. (3) Oref, I.; Tardy, D. C. Chem. Rev. 1990, 94, 1407. (4) Shin, H. K. J . Chem. Phys. 1993, 98, 1964. (5) Anderson, L. A,; Davidsson, J.; Nordholm, S. Chem. Phys. 1993, 174, 111. (6) Bemshtein, V.; Lim, K. F.; Oref, I. J. Phys. Chem., in press. (7) Bemshtein, V.; Oref, I. Chem. Phys. Lett. 1995, 233, 173. (8) Bemshtein, V.; Oref, I., manuscript in preparation. (9) Oref, I. J . Phys. Chem. 1992, 96, 6308. (10) Lim, K. F.; Gilbert, R.G. J . Phys. Chem. 1990, 94, 72. (11) Lim, K. F.; Gilbert, R. G. J . Chem. Phys. 1986, 84, 6124. (12) Gilbert, R. G.; Smith,S . C. Theory of Unimolecular and Recombination Reactions; Blackwell Scientific: Oxford, U.K., 1990. (13) Gilbert, R. G.; Oref, I. J . Phys. Chem. 1991, 95, 5007. (14) Oref, I.; Gilbert, R. S . in: Mode Selective Chemistry; Kluwer Academic Publishers: Netherlands, 1991; p 393. (15) Landau, L. D.; Lifshitz, E. M. Statistical Physics, Part 1; Pergamon Press: Oxford, 1980. (16) Nikitin, E. E. Theory of Elementary Atomic and Molecular Processes in Gases; Clarendon Press: Oxford, U.K., 1974. (17) Lambert, J. D. Vibrational and Rotational Relaxation in Gases, Clarendon Press: Oxford, U.K., 1977. (18) Yardley, J. T. Introduction to Molecular Energy Transfer; Academic Press: New York, 1980; Chapter 4. (19) Bemshtein, V.; Oref, I. J . Phys. Chem. 1993, 97, 1281. (20) Lim, K. F. J . Chem. Phys. 1994, 101, 8756.
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